Additivity and non-additivity for perverse signatures

TL;DR

This paper generalizes Wall's non-additivity theorem to perverse signatures on stratified pseudomanifolds, showing conditions under which Novikov additivity holds or fails, extending classical signature theory to singular spaces.

Contribution

It extends Wall's non-additivity results to perverse signatures on stratified pseudomanifolds, providing conditions for Novikov additivity in singular spaces.

Findings

Proves a non-additivity theorem for perverse signatures.

Identifies conditions under which Novikov additivity applies.

Includes special case for Witt spaces and Siegel's results.

Abstract

A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Wall's non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface,Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegel's version of Novikov additivity for Witt signatures is a special case of this corollary.